The enhancement of the localization length for two interacting particles is vanishingly small in transfer-matrix calculations

Abstract

In response to a recent Comment by Frahm et al. regarding our Letter [Phys. Rev. Lett. {\bf 78}, 515 (1997)], we point out that no ``consistent picture'' exists for the enhancement of the localization length $\lambda_2$ for two interacting particles (TIP) proposed previously by Shepelyansky. In fact there are at least 3 different proposals for the dependence of $\lambda_2$ on interaction and disorder. Most analytical and numerical work following Shepelyansky's original approach neglected the phase correlations inherent in the interference phenomena of localization and thus appears at least questionable. In our Letter, we avoided this problem. Our results based on the transfer matrix method (TMM) led us to ``conclude that the TMM ... measures an enhancement ... which is ... due to the finiteness of the systems ''. In particular, we did not question the results of v. Oppen et al. reproduced in the Comment. We also note that in our Letter we explored the limit of large system size M and not the ``thermodynamic limit''. The latter implies of course a finite particle density quite different from the TIP problem. In any case, $\lambda_2$ does not correspond to ``extended states'', because it remains finite and smaller than M. Finally, to the best of our knowledge, there is no ``scaling theory of localization'' for TIP.